Finite Group Databasebeta

← #323369

Subgroups — #323369

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29219C211yesmaximal, characteristic
3#29219C211yesmaximal, characteristic
4#29218C211yesmaximal, characteristic
5#29206C211yesmaximal, characteristic
6C4×#494C211yesmaximal, characteristic
7#10682C212yesmaximal
8#10681C212yesmaximal
9#9835C212yesmaximal
10#8821C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C4×Q16—22no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2100—22no—
19#2099—22no—
20#2099—22no—
21#2058—22no—
22#1990—22no—
23#1990—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1988—22no—
29#1988—22no—
30#1987—22no—
31#1969—22no—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948—22no—
35#1845C2211yescharacteristic
36#1844C2212yes—
37#1834C2212yes—
38#1834C2212yes—
39#1834C2212yes—
40#1834C2212yes—
41#1834C2212yes—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C22×Q16—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58#494—22no—
59#494Q814yescomplemented
60C4×Q16—48no—
61C2×#107—22no—
62C2×#106—24no—
63C2×C4⋊C8D412yes—
64C2×C4⋊C8D412yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×C4×C8D412yes—
68C2×C4×C8D412yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#384C2312yes—
88#384C2312yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C43C2311yescharacteristic
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×Q8—41no—
110C2×Q16—28no—
111C2×Q16—44no—
112C2×Q16—44no—
113C22×C8C2×D414yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—82no—
129C4⋊Q8C2×Q812yes—
130C4⋊Q8C2×Q812yes—
131C4×Q8—44no—
132C4×Q8—44no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2411yescharacteristic
166#107—44no—
167#106—84no—
168C4⋊C8—24no—
169C4⋊C8—24no—
170#103—44no—
171#103—44no—
172C4×C8—24no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—42no—
180C4×C8C2×Q812yes—
181C4×C8#4112yes—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213Q16—416no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—24no—
224C2×C8—24no—
225C2×C8C22⋊Q814yes—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—82no—
258C42—82no—
259C42#13514yes—
260C42AGL(1,ℤ/8)14yes—
261C42#13714yes—
262C42C22×Q811yescharacteristic
263C42C2×#4111yescharacteristic
264C422−1+411yescharacteristic
265C422−1+411yescharacteristic
266C23C2×#12711yescharacteristic
267Q8—44no—
268Q8—44no—
269Q8—82no—
270Q8—82no—
271Q8—164no—
272Q8—164no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4#49512yes—
298C2×C4#49512yes—
299C2×C4#49612yes—
300C2×C4#49612yes—
301C2×C4#49712yes—
302C2×C4#49712yes—
303C2×C4C2×C22⋊Q811yescharacteristic
304C2×C4#53611yescharacteristic
305C2×C4#54711yescharacteristic
306C2×C4#54711yescharacteristic
307C2×C4D4×Q811yescharacteristic
308C2×C4D4×Q811yescharacteristic
309C2×C4C2×#13512yes—
310C2×C4C2×AGL(1,ℤ/8)12yes—
311C2×C4C2×#13712yes—
312C2×C4#57512yes—
313C2×C4#57712yes—
314C2×C4#57812yes—
315C8—24no—
316C8—44no—
317C8—44no—
318C8—42no—
319C22#254211yescharacteristic
320C22C2×#49511yescharacteristic
321C22C2×#49611yescharacteristic
322C22C2×#49711yescharacteristic
323C22#315311yescharacteristic
324C22#315411yescharacteristic
325C22#315611yescharacteristic
326C4—22no—
327C4—22no—
328C4—22no—
329C4—82no—
330C4—82no—
331C4—162no—
332C4—162no—
333C4—161no—
334C4—161no—
335C4#172014yes—
336C4#307712yes—
337C4#323512yes—
338C4#334212yes—
339C4#335612yes—
340C4#338312yes—
341C4#338712yes—
342C2C2×#172011yescharacteristic
343C2#1963711yescharacteristic
344C2#1968111yescharacteristic
345C2#1970011yescharacteristic
346C2#2852111yescharacteristic
347C2#2853211yescharacteristic
348C2#2854211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
59. #494Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332892
#292192
#292182
#292062
C4×#4942
#106822
#106812
#98352
#88212
#82612